Thanks for the comment. You are right in saying that Grothendieck never used the term “motivic cohomology”. On the other hand, what I meant by (ab)using the term is that it is a functor that takes a smooth proj. scheme and associates to it a pure motive- now this pure motive does have (conjecturally atleast) a realization in terms of “known” cohomology theory. So in this picture, calling the functor motivic cohomology seemed appropriate.

Ofcourse, the term as is used today refers to Bellinson-Voevodsky, … which is that these are simply Ext groups in a (derived triangulated) category of mixed motives, but I guess thats a story for some other day.

Best,

Obi

Now a slight “terminological” complaint: I don’t think the term “motivic cohomology” was ever used by Grothendieck to describe a universal cohomology theory (i.e: the functor you call motivic cohomology) . And the people who did coin it (Beilinson most likely) certainly don’t use it to mean a universal cohomology theory! In their theory/dream/fantasy, motivic cohomology groups are certain ext groups WITHIN an already existing derived category of mixed motives. In particular, motivic cohomology groups have “realisations” simply into abelian groups, and not into things like Galois representations/Hodge structures. A much better name, I think, would be absolute cohomology in analogy with the Galois side.

Sorry to submit an essentially contentless comment, but reading people call motivic cohomology a universal cohomology theory confused me for the longest time before I realised it wasn’t that.